Lecture 4.1: Relations Basics CS 250, Discrete Structures, Fall 2015 Nitesh Saxena Adopted from previous lectures by Cinda Heeren

Course Admin Mid-term 2 HW3 Due Being graded Covers “Induction and Recursion” (Chapter 5) Due 11am Nov 17 (Tuesday) 2/27/2019

Outline Relation Definition and Examples Types of Relations Operations on Relations 2/27/2019

A x B = {<x,y> : xA and yB}. Relations Recall the definition of the Cartesian (Cross) Product: The Cartesian Product of sets A and B, A x B, is the set A x B = {<x,y> : xA and yB}. A relation is just any subset of the CP!! R  AxB Ex: A = students; B = courses. R = {(a,b) | student a is enrolled in class b} We often say: aRb if (a,b) belongs to R 2/27/2019

Relations vs. Functions Recall the definition of a function: f = {<a,b> : b = f(a) , aA and bB} Is every function a relation? Draw venn diagram of cross products, relations, functions Yes, a function is a special kind of relation. 2/27/2019

Properties of Relations Reflexivity: A relation R on AxA is reflexive if for all aA, (a,a) R. Symmetry: A relation R on AxA is symmetric if (a,b)  R implies (b,a)  R . 2/27/2019

Properties of Relations   2/27/2019

Properties of Relations - Techniques How can we check for the reflexive property? Draw a picture of the relation (called a “graph”). Vertex for every element of A Edge for every element of R Now, what’s R? {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} 1 2 3 4 Loops must exist on EVERY vertex. 2/27/2019

Properties of Relations - Techniques How can we check for the symmetric property? Draw a picture of the relation (called a “graph”). Vertex for every element of A Edge for every element of R Now, what’s R? {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} EVERY edge must have a return edge. 1 2 3 4 2/27/2019

Properties of Relations - Techniques How can we check for transitivity? Draw a picture of the relation (called a “graph”). Vertex for every element of A Edge for every element of R Now, what’s R? {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} 1 2 3 4 A “short cut” must be present for EVERY path of length 2. 2/27/2019

Properties of Relations - Techniques How can we check for the anti-symmetric property? Draw a picture of the relation (called a “graph”). Vertex for every element of A Edge for every element of R Now, what’s R? {(1,1),(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,3),(3,4),(4,4)} No edge can have a return edge. 1 2 3 4

Properties of Relations - Examples Let R be a relation on People, R={(x,y): x and y have lived in the same country} 1 ? 1 2 ? 1 2 3 ? Is R transitive? No Is it symmetric? Yes Is it reflexive? Yes Is it anti-symmetric? No 2/27/2019

Properties of Relations - Examples Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Suppose (x,y) and (y,z) are in R. Definition of “divides” Then we can write 3j = (x-y) and 3k = (y-z) Can we say 3m = (x-z)? Is (x,z) in R? Add prev eqn to get: 3j + 3k = (x-y) + (y-z) 3(j + k) = (x-z) Is R transitive? Yes 2/27/2019

Properties of Relations - Techniques Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Is (x,x) in R, for all x? Definition of “divides” Does 3k = (x-x) for some k? Yes, for k=0. Is R transitive? Yes Is it reflexive? Yes 2/27/2019

Properties of Relations - Techniques Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Suppose (x,y) is in R. Definition of “divides” Then 3j = (x-y) for some j. Does 3k = (y-x) for some k? Yes, for k=-j. Is R transitive? Yes Is it symmetric? Yes Is it reflexive? Yes 2/27/2019

Properties of Relations - Techniques Let R be a relation on positive integers, R={(x,y): 3|(x-y)} Suppose (x,y) is in R. Definition of “divides” Then 3j = (x-y) for some j. Does 3k = (y-x) for some k? Yes, for k=-j. Is R transitive? Yes Is it symmetric? Yes Is it reflexive? Yes Is it anti-symmetric? No 2/27/2019

More than one relation Suppose we have 2 relations, R1 and R2, and recall that relations are just sets! So we can take unions, intersections, complements, symmetric differences, etc. There are other things we can do as well… 2/27/2019

More than one relation SR = {(a,c):  bB, (a,b)  R, (b,c)  S} Let R be a relation from A to B (R  AxB), and let S be a relation from B to C (S  BxC). The composition of R and S is the relation from A to C (SR  AxC): SR = {(a,c):  bB, (a,b)  R, (b,c)  S} A B C 1 2 3 4 x y z s t u v R S SR = {(1,u),(1,v),(2,t),(3,t),(4,u)} 2/27/2019

R2 = R1R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)} More than one relation Let R be a relation on A. Inductively define R1 = R Rn+1 = Rn  R A A A R R1 1 1 1 2 2 2 3 3 3 4 4 4 R2 = R1R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1), (4,2)} 2/27/2019

R3 = R2R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)} More than one relation Let R be a relation on A. Inductively define R1 = R Rn+1 = Rn  R A A A R R2 1 1 1 2 2 2 3 3 3 … = R4 = R5 = R6… 4 4 4 R3 = R2R = {(1,1),(1,2),(1,3),(2,3),(3,3),(4,1),(4,2),(4,3)} 2/27/2019

Today’s Reading Rosen 9.1 2/27/2019